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1972 Market Insurance, Self-Insurance, and Self-Protection

Market Insurance, Self-Insurance, and Self-Protection Author(s): Isaac Ehrlich and Gary S. Becker Source: The Journal of Political Economy, Vol. 80, No. 4 (Jul. - Aug., 1972), pp. 623-648 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1829358 Accessed: 08/09/2010 21:54
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Market Insurance, Self-insurance, and Self-Protection

Isaac Ehrlich
University Chicago and Tel-Aviv University of

GaryS. Becker
University Chicago of

The article develops a theoryof demand for insurancethat emphasizes the interaction between market insurance,"self-insurance," The of and "self-protection." effects changesin "prices," income,and othervariableson the demandfor thesealternative formsof insurance are analyzed using the "state preference" approach to behaviorunder uncertainty. Market insurance and self-insurance shownto be subare stitutes,but market insurance and self-protection can be complements.The analysis challengesthe notion that "moral hazard" is an inevitable consequence of market insurance,by showingthat under certainconditions lattermay lead to a reduction the probabilities the in of hazardousevents.

have usuallybeen to The incentive insureand its behavioralimplications analyzed by applyingthe expected utilityapproach withoutreference employedin consumption curve analysis ordinarily to the indifference expectedutility In is theory. thispaper insurance discussedby combining curveanalysiswithinthe contextof the "state prefand an indifference in (the preferences queserence"approachto behaviorunderuncertainty to tion relating statesof the world).' We use this framework restate to
Becker's contributionwas primarilyan unpublished paper that sets out the approach developed here. Ehrlich greatlyextended and applied that approach and was primarilyresponsiblefor writingthis paper. We have had many helpful comments from Harold Demsetz, Jacques Dreze, Jack Hirshleifer, and members of the Labor Workshopat Columbia Universityand the Industrial OrganizationWorkshop at the University Chicago. of 1 An approach originallydevised by Arrow (1963-64) and worked out in application to investment decisionsunderuncertainty Hirshleifer by (1970). 623





and reinterpret a simpler moreintuitive in and way somefamiliar propositionsconcerning insurance behavior;moreimportant, derivea number we of apparently new results, especiallythose concerned with self-insurance and self-protection. Our approach separates objective opportunities from"taste" and otherenvironmental factors, whichfacilitates indean pendentinvestigation each class of factorsanalyticallyas well as of In empirically. addition,we considernot only the incentiveto insure, but also how muchinsurance purchased is undervarying "opportunities"2 and in viewof the existence the alternatives self-insurance selfof of and protection. use the basic analyticaltools employedthroughout We traditionalconsumption production and theory. It has been argued that insuranceis different from"ordinary"goods and services because it is not desiredper se, but as a meansof satisfying morebasic needs.3Recent developments consumption in theory4 suggest, however, thatthe distinction between goodsand services purchased the in market and morebasic needstheysatisfy not a uniquecharacteristic is of insurance, but applies to all goods and services.The demand for the latteris also derivedfrom needs theysatisfy, the just as the demandfor factors production ordinary of in production theory derivedfrom is their contribution finalproducts. to The basic needsunderlying purchaseof insurance the will be identified withconsumption opportunities contingent upon the occurrence various of mutually exclusiveand jointlyexhaustive "states of the world."5 Market insurance this approachredistributes in incomeand, consequently, consumption opportunities, toward less well-endowed the states.Self-insurance, however,redistributes income similarly,self-protection a related has and effect, either might pursuedwhenmarket be insurance was not available. Moreover, optimaldecisionsabout market insurance dependon the availability theseotheractivities of and shouldbe viewedwithin conthe textof a morecomprehensive "insurance" decision.
2 Theorems concerning optimal insurancedecisions have been derived in two recent contributions Smith (1968) and Mossin (1968). Our approach differs by not only in formbut also in substance; for example, in the analysis of the interactionbetween marketinsurance, self-insurance, and self-protection. 3 For example, Arrow (1965) says, "Insurance is not a material good . . . its value to the buyer is clearly different kind from the satisfactionof consumer'sdesires in for medical treatmentor transportation. Indeed, unlike goods and services, transactions involving insurance are an exchange of money for money, not money for somethingwhich directlymeets needs" (p. 45). 4 See, for example,Becker and Michael (1970). 5 By consumption opportunities each state of the world is meant command over in commodities, producedby combining Ci, marketgoods, Xi, time spent in consumption, E ti, and the "state environment," , via householdproductionfunctions(for the latter conceptsee Becker and Michael 1970): Cij = fij(Xij, tij, Ei) j - 1, . . . , m where i refers different to commodities. the productionfunctions If fullyincorporatethe effects of environment, utilityfunctionof commodities the would not depend on which state occurred. In particular,for an aggregate commodityC, U(C0) = U(C1) if C0 = 1' where 0, 1 denote different states.





The first partof thispaper spellsout a modelof market insurance and discussesthe effects changesin termsof trade,"income,"and other of environmental factors on optimal insurance decisions. Self-insurance, and self-protection, a simultaneous determination the full insurance of decisionare thendiscussedin the second and moreoriginalpart.
I. Market Insurance

We assumeforsimplicity thatan individual facedwithonly two states is of theworld(0, 1) withprobabilities and 1 - p, respectively, that p and his real incomeendowment each state is given with certainty IUe in by and 1,", where I," - It,' is the prospective loss if state 0 occurs. If income in state1 can be exchanged income state0 at thefixed for in rate


state 1. The amountof insurancepurchasedin state 0 can be defined as the difference betweenthe actual and endowedincomes:6
s - Io - Io". (2)

it can be called the "price of insurance" measured termsof incomein in

The expenditure insurance on measured terms state l's incomeis in of b -Ile

_I, -Sjt.

(3 )

Substituting in (3) givestheopportunity (2) boundary

or the line 21Bin figure It is assumedthat the individualchoosesthe 1.7 the optimalincomein states 1 and 0 by maximizing expectedutilityof the incomeprospect,


p) U(1i) +

p U(Io),8


The firstsubject to the constraint givenby the opportunity boundary. orderoptimality condition is p_ UO' (1



6" Note that insuranceis definednot in termsof the liability"coverage" of potential losses, as in Smith's (1968) and Mossin's (1968) papers, but in terms of "coverage minuspremium," the net addition to income in state 0. or 7 In figure1, the opportunity boundaryAB is drawn as a straightline. This assumes that the same termsof trade apply to both insuranceand "gambling,"that is, to movements to the right and to the left of E, the endowment position. In practice, the opportunity boundary may be kinked about the endowment point. 8 For analytical simplicity we ignore the time and environment inputs and assume only a single aggregatecommodityin each state. Then the output of commoditiescan be identified with the input of goods and services,or with income.







Certainty Line






~ ~~~~~~~B
FIG. 1




where(pU0') (1 -p) U1' is the slope of the indifference curve (defined alongdU* 0), and n is theslopeof thebudgetline.In equilibrium, they mustbe thesame (see pointP). One can more completely factors separate tastes fromenvironmental by dividing (1 - p) through (6) to obtain p in

1p p


U0' .



Further,i, the price of insurancedeflatedby the actuarially"fair"9 because a of price,pl(1 - p), is a measure the "real" priceof insurance fairpriceis "costless"to theindividual(see the secondparagraph below). the Equation (7) thusimpliesthat,in equilibrium, real priceof insurance equals the ratioof the marginal utilityof I,) to that of I,, the ordinary demandtheory. resultin consumer curvebe conthat the indifference condition The second-order requires at vex to the origin the equilibrium point,or D -_p


2 (1



> O.


A sufficient conditionis that the marginalutilityof incomeis strictly An immediate of implication equation (7) is that insurancewould be

9 An actuariallyfairexchangeis an exchangeof p/(1 - p) units of income in state 1 for an additional unit of income in state 0, where p/(1 - p) is the odds that state 0 would occur. 10Hirshleifer marginalutilityof (1970, p. 233) points out that althoughdiminishing income is not a necessaryconditionfor equilibrumat any given point,it is a necessary curve to be convex at all points. conditionfor the indifference





demanded-some I, would be traded forIo if the slope of the indifference curveexceededthe priceof insurance theendowment at point,E:


U (I,.)


If the opposite were true, "gambling"would be demanded,provided similarterms tradeapply in redistributions incometowardstate 1. of of Note that gamblingcan occur withoutincreasingmarginalutility of incomeif the opportunities available are sufficiently favorable. Therefore, inferences about attitudestowardrisk cannotbe made independently of existing market opportunities: person a may appear to be a "riskavoider" underone combination pricesand potential of losses and a "risk taker" underanother." If the price of insurancewere actuariallyfair, equation (7) would reduceto 1 - U(' U,': incomes wouldbe equalized in both statesof the world if the marginalutilityof incomewere always diminishing. This is "full insurance"in the sense thata personwould be indifferent to as whichstate occurred.'2 particular, small changesaroundthe equiIn for librium position, would act as if he were indifferent he towardrisk and interested onlyin maximizing expectedincome.Indeed,his incomein his 13 each state would equal his expectedincome; therefore, insurance fair can be regarded costlessto him.'4 as
11 Indeed, when faced with several independenthazards, a person might "gamble" and "insure" at the same time,provided the different hazards were associated with differentopportunities. For example, given a fair price of theftinsurance,he may fully insure his household against theftand at the same time engage in a riskyactivityif his expected earningsthere were greaterthan his earningsin alternative"safe" activities (see Ehrlich 1970). 12 Full insurance can be identified with full coverage of potential losses, since the equation I, I=) impliesthat I1e _b = 1se + d - b, where d is the grosscoverageand e- Isle.By the same reasoning, b is the premium. Clearly,then,d = since an "unfair" price of insurance > 1 implies that I1 > Io, it also implies necessarily less than full coverage of potentiallosses. 13If I,0 I1=zI that is, I= Iie-s where Az=p/(1-p), then =Io=Ioe+s, 14 ( 1-p) Ile.

Althoughthe model has been developed for two states of the world, the analysis applies equally well to n states. We define the state with the highest income-say, state n-as the state without hazard and define all the states with hazard (h = 1, n - 1) relativeto that state. Denoting by Ph the probabilityof state I, by

I = pIO e+



h= 1

the probabilityof state n, and by Th the implicitterms of trade between income in state n and income in state h, it can easily be shown that if the termsof trade were fair (Jnh [P/P,] rsh= 1) Sh would be chosen to equalize incomes in all states of the = world and losses would be "fully covered." If the real termsof trade were unfairbut





A. Substitution Effects The effect an exogenousincreasein the price of insuranceon the of demandfor Io, with the probability loss and the initial endowment of being the same, can be foundby partiallydifferentiating first-order the optimality condition withrespectto t: 010


1 = Dj [(1

P) Ui' + (Io


r (1 -P)



the Since the denominator has alreadybeen shownto be positive, sign D or of equation (10) is the same as the signof the numerator, negativeif in Io > Io', since we are assumingU1" < 0. An increase the relativecost of incomein state 0 necessarily decreasesthe demandforincomein this state. Moreover, also reducesthe amountof insurance it purchased, since unchanged: Ioe remains aslan - aI(l)a - ajoean - ajola. Similarly, effect an increasein 3 on I,, and thuson the amount the of spenton insurance, is 011

1 D [1

_ p) Ulf + (Io oe) p Uoff
Joe) pUo" - spUO"


Here theresult ambiguous is sinceUl' is positive whereas

is negativeif Uo" < 0 and s > 0. The resultis ambiguousbecause, although increasein n reducesthe amountof insurance an purchased, each unitpurchased becomesmoreexpensive. Consequently, amountspent the on insurance woulddeclineonlyif thepriceelasticity demandforinsurof ance exceeded unity15 proof obvious). (a is effect Equations (10) and (11) do not isolate a "pure" substitution because an increasein n lowersthe opportunities available (if s > 0). If bothI, and I( are superior effects goods, the incomeand substitution on bothreducethe demandforI,, whereas theyhave oppositeeffects the demandfor1I. Diagrammatically, the opportunity as boundarychanges P AB from to CD (see fig.2), the equilibrium pointshifts from to Q. If 1o werea superior good, Q mustbe to theleftof P'. Even if Io werean ininferior good, however, "pure" (that is, expenditure-compensated) a crease in the termsof trademustalways reduce the demandforIo and like S. increase theequilibrium mustshift from to a pointto its left, P II:
constant (7th = 1 + X> 1 for all h), sh would be chosen to equalize incomes in states with hazard only, that is, we would achieve what has been called full insurance above a deductible (for a definitionof this concept and an alternative proof see Arrow 1963). 15 This analysis,therefore, also shows that the effect a change in n on the "fullof ness" of insurance (the difference - Io) and thus on the degree of gross coverage I is generallynot unambiguous.






P 0 D
FIG. 2



Income Effects

Equation (4) can be written as
Ile + ?IoeW _I, ? + Io, (12)

whereW is a measureof the totalopportunities available. (This is shown in fig. 1 by the intercept on the I, axis.) The effect a changein OA of theendowments theincomedemanded each state can be determined on in by differentiating first-order the condition:

aIo aW


aIo aIe

ajoe -aie






aIl adoi

ail" aw

aIl aIoe



whereD;q _r(1 - p) U1", and D-)2 pUJ". The incomedemandedin each state necessarily if increaseswithopportunities the marginal utility of incomeis falling.Hence, an increasein each state's endowment increasesthe demandforincomein otherstatesas well. The effects the on demandforinsurance morecomplicated, are however, since theydepend on how different if endowments change.For example, Ile alone increased,


at1 i







if and the demandforinsurance would increase.Similarly, Joe alone increased,


1< 0



and the demandforinsurance would decrease.Equations (14) and (15) implythat if the difference endowedincome-the endowedloss from in a demandforinsurance would increase.Put differently,personwould be morelikelyto insurelarge ratherthan small losses (see Lees and Rice on of 1965).17 The effects a change in total opportunities the demand for insurancecannot be derivedwithoutknowledge the way opporof is tunities changeessentially because insurance a "residual" that bridges states the gap betweenendowedand desiredlevels of incomein different of the world.18 (and hence the size of the loss) are For example, both endowments if changed thesame percentage, by then

the hazard

increased either because Joe decreased or Ile increased, the






elasticities demandfors and I,, respectively.'9 of Equation (16) incorporates the ratherobvious conclusionthat the on depends effect a changein opportunities the demandforinsurance of on the effects the incomedemanded each state.If the slopes of the in on indifference curvesare constantalong a given ray fromthe origin (the indifference curvesare like EPF and GQ1H in fig.3)-there is constant
16 According to equation (13), aI,/OI()e(1 - p) U1,,]/[_pU("tt -;2 (1 [-.2 p) Ui"] = 6, where clearly 0 < 6 < 1 if Ul" and U(" < 0. But since s - Jo- Ioe,

where E,1w asl0W - Ws and

o -Iol&W

* WlIo are the opportunity

17 Similarly,he would be less likely to take large gambles (see the discussion in Hirshleifer 1966). Of course,if insuranceis fair he will fullyinsure all losses, large or small. 18Note the analogy between insuranceand savings: the latter bridges the gap beat points in time. tween "endowed" and desiredlevels of consumption different 19 Given s -_ and I1 C y 0I(e, then (Os/OW)(W/s) = (ds/dI(C) (I(C/s) = I(-it)e 1 no (o/s) - (Ioe/s); by collectingterms,we get equation (16). Since IO s, eW if al > 1. L that is, if 10e = Ile If the loss is unaffected an equal increasein endowments, by where L is a constant,then se = (ds/dIle) (Ile/s) = (10/s)PT (d log W) /(d logIle) _(I1e/s). This impliesthat




1 <0.

> e=



> ro=
< Io(I


> 1

(if Ii > lo)o








B FIG. 3



positionslie on a givenray relative riskaversion20 thenall equilibrium - 1. An equal 3, from origin, P and Qi do in figure and elf the as would thenincreasethe demand proportional increasein all endowments for insurance the same proportion. the slopes of the indifference If by

Note that

d slope

d do1

pUO p )u1'
y -(UO"IUO');

subject to Ii = yIO is

] J
the latter definesin-

0 as -(U199/Ul')

constant, decreasingrelativerisk aversion.Similarly cr creasing,

d d1o
subject to I10


L (1 - p) U1' -


Lis L

0 as - (U199/U19) -(UO"I/UO');

the latter definesin-

creasing,constant,or decreasingabsolute risk aversion (see Pratt 1964, Arrow 1965). (Diagramatically,constant absolute risk aversion implies that the slopes of the indifference curves are constant along any 450 line joining two equilibriumpositionscurves are like EPF and IQJ in fig. 3.) Equation (16a) in n.19 the indifference implies that increasingrelative risk aversion, rO > 1, is compatible with decreasing absolute risk aversion,ese < 0, only if o < W/IO(l + Jc).





and o and thus is relativeriskaversion, IQ2J in figure there increasing 3, relativeriskaversionimpliesthat the Fsw wouldexceedunity.Increasing

curvesincreasealong a givenray from origin, shownby EPF and the as

between and I, tendsto declineas opportunelasticity substitution of 1,) in elsewhere the ities increase.21 Regardlessof the shape of preferences relative(and absolute) risk aversionremains preference space, however, line. This constancyalways characterizes constantalong the certainty choiceswhenthe priceof insurance actuarially is fair (see fig.4). C. Rare Losses givenin equation conditions insurance for of An inspection the necessary of (9) shows that changesin p, the probability loss, do not affect the is incentive insureas longas the real priceof insurance independent to of p. If insurancewere actuariallyfair,the real price would always equal of unity, and thuswouldbe independent p. The deviationfroma fairprice,or the "loading" in insurancetermifrom identity the nology, be defined can

(1+-FP'(17) )(17) -





Increasing relative
risk aversion


Certainty Line: constant relative and absolute risk aversion





21 Since the slopes of the indifference curves necessarilyare constant along the "certaintyline" and by assumptionbecome increasingly steep toward I along other rays fromthe origin,a given percentagedeviation of the price of insurancefrom the fairprice resultsin smallerpercentagechangesin the ratio I1/I0 at higherindifference levels. That is, o-= (dlogIl/Io)/dlog t decreases at higher indifference levels when nt equals the fair price.




where is the "loadingfactor." X wereindependent p, so also would X If of be the real priceof insurance and p wouldhave no effect the incentive on to insure.In particular, therewould not thenbe a greaterincentive to insure"rare"lossesof a givensize.22 Since, apparently, rare losses are more frequently X insured,23 is presumablypositively relatedto p, perhapsbecause processing and investigatingcosts increaseas p increases.24 (An alternative is explanation provided by the interaction betweenmarketand self-insurance analyzed in the nextsection.) Even if the incentive insurewereindependent p, to of theamount insured woulddeclineand the expenditure insurance on would increase p increased.25 as II. Self-Insurance and Self-Protection

Two alternatives marketinsurancethat have not been systematically to analyzed in the literature insuranceare self-insurance-areduction on in the size of a loss-and self-protection-a reduction the probability in of a loss.26 For example,sprinkler systemsreduce the loss fromfires; burglaralarms reduce the probability illegal entry; cash balances of reducefluctuations consumption; in medicines, certainfoods,and medical
22 This result appears to contradict one by Lees and Rice (1965) because they definethe loading factorin termsof the gross rather than net amount paid in claim; that is by X' in Jt= [(1 + X') p]/[1 - (1 + X') p]. A reductionin p, ' held constant, would reduce 2,-our definitionof the loading factor-and thus would increase the incentiveto insure. 23 Some evidence is presentedin Lees and Rice (1965). 24 Let the amount a that is spent processing and investigating each claim be the only administrative cost of providinginsurance.In a zero profitequilibriumposition, the unit price of insurancewould equal the ratio of the total amount collectedin premiums in state 1 (including administration costs) to the difference between the net amount paid in claims in state 0 and administration costs: A = (p d + p a)/[d (1 - p) -p a], where d is the amount covered by insurance.The degree of loading defined by X = = (d + a)/[d-p ]-1 = a/[d-p (d + a)] would be [(1-p)/p1 ;t-1 a/(l-p) larger the largerp was if d were fixed (d would tend to decrease as p increased,and this would increase ?i even further). 25 Generally, the effect an increasein p on the optimal values of If) and I , assumof ing that t = [(1 + X) p]/(i - p) and that i, le, and Ie are constant,is given by







D lLoS




p i<


provided U" < 0. An increasein p would then lower the optimal amount of insurance s = IO - Ise and increase the optimal expenditureon insurance b -- 11e - I1. 26 These have been called "loss protection"and "loss prevention," respectively(see Mehr and Commack 1966, pp. 28-29).





checkupsreducevulnerability illness; and good lawyersreduce both to and the punishment crime. As these for the probability conviction of artificial distinguish to examplesindicate,it is somewhat behaviorthat reducesthe probability a loss from of behaviorthat reducesthe size of a loss, since many actionsdo both. Nevertheless, do so for expository we the insurance convenience and because self-insurance clearlyillustrates principle redistributing of incometowardless favorablestates.
A. Self-Insurance

person as L

Assume that marketinsuranceis unavailable and write the loss to a - L(Lc, c), where Le - I1, - IoC is the endowed loss, c is -L'(c) < 0. The expected

the expenditure self-insurance, on and can utility be written as
U* ( 1 -p) U(Ile C) +P




The value of c that maximizes equation (18), c0, satisfies first-order the condition L'(cO) + 1 (1 -p) U' *(19)

This maximizes expected utility the marginal if utility incomeand the of marginalproductivity self-insurance decreasing, of are that is, if the indifference curvesare convexand if the production transformation curve between income states 1 and 0 (TN in fig.5) is concaveto theorigin.28 in A necessary condition a positiveamountof self-insurance for obviously is -L'(cO) > 1, or that therebe a net additionto incomein state 0. A sufficient if condition, the transformation indifference and curvesdo not have kinks,is that
L'(Le, 0) + p U' (Io) 29




p) U' (fle)


An increase the unitcost of self-insurance, in measured the marginal by
27 For analyticalconveniencewe assume that Ic alone is affected by c, although,of course,both endowmentsmay be affected. Moreover, the assumptionthat aL/ac < 0 is not always true: an individual could increaseIle and reduce If, by deliberately exposing himselfto hazards; for example,by committing crime or engagingin a risky a legal occupation (see Ehrlich 1970). The condition aL/ac > 0 can be said to define "negativeself-insurance." 28 See equation (A5). Note that the transformation curve may be kinked at the endowmentpoint. 29 If the opposite were true, there would be an incentive to increase the loss by increasing and reducingI0 (see n. 27 above). I,




A T Certainty Line E M





N 5


wouldreducethedemandforself-insurance, of productivity self-insurance, by measured CO:30 ac0


wherea is a parameter that reducesthe absolutevalue of L' fora given a c. Similarly, reductionin Ifc would increase the demand for selfinsurance:

unlikethe to Equation (20) showsclearlythattheincentive self-insure, to is incentive use marketinsurance, smallerforrare losses. The reason >0.32 is thattheloadingfactor self-insurance largerforrarelosses because of is
30 Althoughc denotes the expenditureon self-insurance rather than the reduction in the size of the loss, there is a one-to-one relationshipbetween expenditureand insurancebecause -L'(c) > 1. 31 By differentiating equation (19) with respectto ca-p, I(e and Ile held constant< one obtains ac(l/da -= (pU('/U*( (() (OL'/Oa) (+)/(-) < 0, where U*(.(.- 2U*IaC2 < 0 (see AppendixA), and by assumptionaL'/ac > 0. 32By differentiating equation (19) with respectto I( - ee, p and L' held constant, one obtains -ac0/d0Ioc= acOl/Lc -{pUo"[L'(c) + 1]/U*,,} aL/aLe >0, where by assumptionaL/OLe > 0.





An increase endowed in in incomes thatresulted from investment human capital would probablybe associated with an increasein the marginal productivity self-insurance.34 of on Therefore, effect self-insurance the of a changein incomehas to be separatedfromthe effect the associated of changein marginal productivity. If marketand self-insurance were both available, values of c and s wouldbe chosensimultaneously maximize expected to the utility function,
U* - (1

independenttheprobability loss.33 of of

its price,unlike the price of marketinsurance, to can be presumed be

p) U(Ile




s n) + p U(ll




c + s).

(23) If the priceof market insurance wereindependent the amountof selfof insurance, first-order the optimality conditions wouldbe




+ 1] O.


p) U1' - p Uo'[L'(c)

By combining theseequationswe get L'(c) +1


In equilibrium, therefore, "shadowprice"of self-insurance the wouldequal the priceof market insurance. Clearly,marketinsuranceand self-insurance "substitutes" the are in sensethat an increasein x, the probability loss beingthe same,would of decreasethe demandformarketinsuranceand increasethe demandfor self-insurance.35 example, changein the market For a insurance line from AB to CD in figure5 would increaseself-insurance the horizontal by distancebetween and M2 and reducemarket M1 insurance thehorizonby tal distance between and P. In particular, purchase market Q the of insurance wouldreducethe demandforself-insurance-compare pointsS and, say, M1. Whenmarket insurance availableat a fairprice,theequilibrium is condition (25) becomes 1 P I-p ' I L'(c) + or (26)
33 The price of self-insurance given by i = - 1/[L'(c) + 1], where L'(c) preis sumably does not depend on p. The loading factoris then given by X =- {1/[L'(c) + 1]} (1 - p)/p] - 1. Hence A/ap <0. 34 That is, not only would li~e/aE 0 i = 0, 1 whereE is the stock of human capi> tal, but probably also 02L/OcOE< 0. 35 A mathematical proof can be found in AppendixA.



LI'(c) -_-,

preciselythe conditionthat maximizesexpected income.36 Even with diminishing marginal utility income, personwould act as if he were of a riskneutraland choose the amountof self-insurance maximized that his expectedincome. Consequently, apparentattitudestowardrisk are dependenton marketopportunities, and real attitudescannot easily be inferred from behavior. More generally, evenif the priceof market insurance werenot fair,the optimalamountof self-insurance would maximizethe marketvalue of income (given by W in equation [12]), and would not depend on the shape of the indifference curvesor even on the probability distribution of Geometrically, optimalself-insurance determined moving states.37 is by alongthetransformation in figure to thepointof tangency curve 5 between thiscurveand a market insurance line; since the market value of income is the intercept the y-axis,thatintercept on wouldbe maximized such at pointof tangency. The effects specific of parameters the demandformarketand selfon insurancewhen both are available oftenare quite different fromtheir effect when marketinsuranceor self-insurance alone is available. For example,although increasein the endowedloss increasesthe demand an forself-or marketinsurance wheneitheralone is available since an increasein market insurance itself reduces self-insurance, vice versa,the and indirect effects can offset directeffects the whenboth marketand selfinsurance positive(see Appendix foran exampleof this). Similarly, are A becausea decreasein theprobability loss withno changein the market of loading factorreduces the demand for self-insurance, increases the it demandformarketinsurance. Therefore, people may be more likely to use the market insurerare losses not necessarily to because of a positive relationbetweenthe probability loss and the loadingfactor(see the of in discussion SectionIC), but because of a substitution betweenmarket and self-insurance. B. SubjectiveProbabilities, Self-Protection, and "Moral Hazard"

Self-insurance and market insuranceboth redistribute income toward hazardousstates,whereas self-protection reducesthe probabilities these of states.Unlikeinsurance, self-protection not redistribute does income, because theamount spentreducing probability a loss decreases the of income
+j p [Ic

36Equation (26) can be derived by maximizing(1 - p) (Ilc _ c) - L(LC, c) - c] with respectto c. 37 Equation (25) can be derived by maximizingW= (Ie - c) + IC - L(L', c) c] with respectto c. We are indebtedto Jacques Dreze for emphasizingthis point.





in all statesequally,leavingunchanged absolutesize of the loss (its the relativesize actuallyincreases). Studiesusingthe states-of-the-world approachto analyzedecisionmaking under uncertainty have assumed that the probability a state is of entirely determined "nature" and is independent human actions. by of Withthisapproachthereis no such thing self-protection; activities as the we call by thisnamewouldbe subsumed underself-insurance. has been It claimedthat statescan always be defined guarantee independence to the of theirprobabilities from humanactions,38 we denythat thiscan be but done in a meaningful way. Consider, example,the probability for that a givenhousewill be damagedby lightning.39 Since thisprobability be can reducedby the installation lightning of rods,independent state probabilities could be obtainedonly by usinga morefundamental state description: the probability a strokeof lightning of itself.If controlof the is weather ruled out, the probability lightning be assumedto be of can unaffected humanactions.We are concerned, by however, about the probabilityof damage to the house we do not care about the probability of lightning se and the probability damage is affected lightning per of by rods. In other words,althoughan appropriatedefinition states would of produce state probabilities that are independent human actions, it of would not producea probability distribution outcomes-the relevant of probability distributionthat is independent theseactions.Since one of of the main purposesof the state-of-the-world approachis to equate the probability distribution outcomeswith the probability of distribution of states, a search for state probabilities of that are independent human actionswouldbe self-defeating. To look at the difference betweenself-protection self-insurance and fromthe viewpoint outcomes, of assume the probability distribution of endowed outcomes givenby AB in figure Self-insurance, contracting 6. by the distribution say, CD, lowersthe probability both highand low to, of outcomes,therebyunambiguously reducingthe dispersion outcomes. of on Self-protection, the otherhand, by shifting whole distribution the to the left to, say, EF, reducesthe probability low outcomesand raises of the probability highones and does not have an unambiguous of effect on the dispersion.40 Since the preceding discussionshows that self-insurance to be disis from tinguished self-protection, developa formal we analysisof the latter. Let us assumethattheprobability a hazardousstatecan be reduced of by
:"'3This example is discussedby Hirshleifer (1970). 40)The effectof the introductionof self-protection on the variance of income, I(C) 92, can be found by differentiation v'(r) =-a Var(Ic)/ar - (1 (Ii p(O-p)

38The only explicitdiscussionis by Hirshleifer (1970, p. 217).







D 6



ity of hazard,r is the expenditure self-protection, plar on and p'(r) 0. If no market self-insurance or wereavailable,the optimalexpendi<, tureon self-protection would maximize

appropriate expenditure: p -

p (p0, r), where pe is the endowed probabil-

[1 -p(pe, r)] U(Ie

r) + p(pe, r) U(Ioe



theoptimality condition is

p'(rO) (U1





p) U1' +p Uo'.


The term theleftis themarginal on gain from reduction p; thaton in the the right, declinein utility the due to the declinein both incomes, the is marginal cost. In equilibrium, course,theymustbe equal. of The second-order optimality condition requires that

p"(rO) (U1


Uo) + 2p'(rO) (Ul'



+ (1-P)



Decreasingmarginal utility incomeis neither necessary of a nor a sufficient condition. p"(rO) > 0, equation (29) is always satisfied the If if marginal utility incomeis constant of and may or may not be satisfied if the marginal utilityis decreasing increasing. or This shows that the incentiveto self-protect, unlikethe incentive insure, not so dependent to is
2p) (11_CI0e)2p'(0),
as p 1/2.

where r is the expenditure self-protection. on Clearly v'(r) -0





on attitudes towardrisk,and could be as strong riskpreferrers for for as riskavoiders. As withmarket and self-insurance, effect a changein incomes the of on thedemandforself-protection dependson thesourceof thechangeas well as on preferences.41declinein lIo alone might A not increasethe demand for self-protection, if the marginalutilityof incomewere falling, even because a declinein Io" would increasethe marginal cost of self-protecA declinein the marginal productivity self-protection-an of increase in the shadowpriceof protection-always decreasesthe demandforselfprotection regardless attitudes of towardrisk.43 Therefore, the endowed if probabilities incomeswerethe same, moreefficient and providers selfof protection would have lowerequilibrium probabilities hazard. Conseof quently, different persons different use probabilities theirdecision-makin ingprocess onlybecauseof differences "temperament," optimism, not in or but also because of differences productivity self-protection. sugin at As gested in the last section,differences productivity, turn,may be in in attributed differences education to in and otherforms "humancapital." of If marketinsuranceand self-protection were jointly available, the function
[1 tion.42

_p (pe, r) ] U (I,,,

r -s 7t(r)) + p(pc, r) U (Iot- r +s) (30)

would be maximized with respectto r and s; the first-order optimality conditions are -(1 -p) Ul + p UO' (31)
- p'(r*)(U,Uo)

(1 - p) Ul'[1 +s*z i'(r*)


p U'0 -O.


The termit'(r*) measures effect a changein self-protection the the of on priceof market insurance through effects p and the loadingfactor its on A.Fromthe definition n in equation (17) we obtain: of
-r '(a) ='(p) p'(r) + ' (r). (33)

41 An equal proportional increasein endowments(Ile = yI0e) would increasethe demand for self-protection (dr0/dIOC)= (1/U*,.) [p'(r ) (U1'y - U(') if conditionif U" < 0 is (U1'/U(') (II,,/Ie) + (1 - p) U1" y + p U0s"]> 0. A sufficient ) 1, or that the "average relative risk aversion" between Ioe and 11C be sufficiently greaterthan one. 42

That is, -ar(I/OId = [p'(rO) U 0


0 as-[p'(rO) (?)/(-)


-U. U/U ' 43 That is, rO0/la [(U1 sumption ap'/ap < 0.

UO)/U*,..] (Op'/af3)-

> 0, where by as-



64 I

The effect marketinsurance the demandfor self-protection of on has moralhazard refers generally been called "moralhazard." In particular, to an alleged deterrent effect marketinsuranceon self-protection44 of thatincreases actual probabilities hazardousevents(Arrow1962). the of Consequently, moralhazard is said to be "a relevantcost of producing insurancethat is imposed by the insuredon the insurancecompany" of (Demsetz 1969,p. 7) and to providea "limitto the possibilities insurance" (Arrow1962, p. 612). We showedin the last sectionthat market insurancedoes reduce self-insurance, no one has shown rigorously but insurance reducesself-protection. why,or underwhat conditions, market on On Market insurancehas two oppositeeffects self-protection. the one hand, self-protection discouragedbecause its marginalgain is is betweenthe incomesand thus reducedby the reduction the difference of the utilities different in states (see equation [28]); on the otherhand, it is relatedto the is encouraged the priceof market if insurance negatively of amountspent on protection through the effect these expenditures on theprobabilities. of Consider relative the importance theseoppositeeffects in two extreme cases: If market insurance were always available at an actuariallyfairprice then p/(1 -p), regardless the amountspenton self-protection, at of and equation (31) impliesthat the optimalamountof marketinsurance (s*) equalizes incomein bothstatesof the world.There is still an incenbecause ;r is negatively related tive to spend on self-protection, however, to theseexpenditures (r):

t (

( ) (1p)2 2.


Substituting U1 U) and Ui' tion (33a) and the factthat5p'(r*)


U0 into equation (32), and usingequa(1 - p) (Ic I(- J), we get 1 (34)

'-p _*

precisely condition maximize the to expectedincome.As withself-insuran on ance, a fairpriceof market insurance encourages expenditure selfmoral hazard protection that maximizes expectedincome.Consequently, wouldnot thenincreasethe real cost of insurance, reducean economy's of insurance since an technical efficiency, limitthe development market or
44 See, for example, Arrow (1962, pp. 612, 613, 616; 1963, pp. 945, 961). Some writershave viewed moral hazard, in part, as a moral phenomenonrelated to fraud in the collection of benefits(see, for example, Mehr and Commack 1966, p. 174): a fireinsurancepolicy, for example, may create an incentivefor arson as well as for carelessness. Our analysis deals explicitlyonly with the effects marketinsuranceon of self-protection, although implicitlyit applies also to the effects fraud. on





amountof marketinsuranceequalizing income in all states would be chosen. Even moreimportant that,contrary the moralhazard argument, is to the optimal expenditure self-protection, can be larger than the on r*, amountspentin the absenceof market insurance, By equations (34) r0. and (28), and thecondition p'(r) < 0, r* wouldbe larger thanr0 if
U (e-

r0) -U



< (1

p) U'(Ile-

r) + p U'(Ioe




whichis likelyprovided is not verysmall and U is concave.Indeed, if p r* utility werea quadraticfunction income, wouldbe largerthanrt if of p were largerthan one-half.45 Not only are marketinsuranceand selfprotection complements the sense that the availability the former in of could increasethe demandfor the latter,but also in the sense that an increase theproductivity self-protection a decreasein the real cost in of or of market insurance would increasethe demandforboth (see Appendix B). that the priceof marketinsurance Suppose,at the otherextreme, was independent expenditures self-protection of on the loading factorincreased sufficiently offset to exactlythe reduction the probability in of loss. Self-protection would thenusuallybe discouraged marketinsurby ance moralhazard would exist because the main effect introducing of market insurance wouldbe to narrowthe differences betweenincomesin different states.46 Moreover,since the demand for marketinsuranceis negatively relatedto the degreeof loading,it wouldbe negatively related to expenditures self-protection. on for Consequently, thosekindsof market insurance withpricesthatare largely independent expenditures selfof on protection, shouldobserveeithera largedemandforinsurance one and a small demandforself-protection, the converse. our judgment, or In this
45If U= aI+ b2, with b <0, equation (35) becomes [p- (1/2)] Ie_ [p_ (1/2)] I(" > 0. Since Iie > I (6, thisimpliesthat p > 1/2. 46 If ;t'(r) = 0, the optimality conditionfor r, given the value of s, is fromequation (32): -p'(r*) [U(I1 -r* - sT) - U(IC - r* + s) ]-(1-p) U' (I c r* -s Js) - p U' (Ioe - r* + s) 0. Self-protection would be discouragedby market insurance if an exogenousincreasein the latteralways reducedthe optimal value of r*; that is, if dr*/ds< 0, or dr*/ds= {p'(r*) [U1' (-3T) - UO'] + [(1 - p) U1" (-Js) + pU0"1 }/ U*rr< 0, where U*rr< 0. The first termin the numerator necessarily is positive since p' (r*) is negativeand r, U0' and U1' are all positive. Therefore, sufficient a condition for the inequalityto hold is that the second term be nonnegative, since r = [(1 + or A) p]/(l - p), that U(" > (1 + X) U1"'. If X > 0-no negativeloading-this latterinequality necessarilyholds provided U" and U"' < 0; for example, if U were the quadratic function I + b 12, with b < 0. Of course,it mighthold even if U "' > 0. a




is the majorreasonwhycertainkindsof hazards,like failure business, in are notconsidered insurable themarket. by Since the price of self-insurance independent the probability is of of hazard (see the discussion Section11A) and thus of expenditures on in our insurance thatself-insurance self-protection, analysisof market implies is likelyto createa moralhazard.That is to say, the availability selfof insurancewould discourageself-protection and vice versa. Moreover, in technological of progress theprovision one wouldtendto discourage the other. This analysisof moralhazard applies not only to the relationbetween self-protection insuranceas ordinarily and conceived,but also to the relationbetweenprotection and insurancefor all uncertain eventsthat can be influenced humanactions.For example, unemployment by comdo pensation, relief, negative or incometax ratesincrease probability that the someone becomesunemployed? Does the presence underground of shelters increasethe probability that a country goes to war, the use of seat belts the probability an automobile of accident,or generous parentalsupport the probability that children become "irresponsible"? Since each of these,in effect, to of relatesa form insurance a form of protection, answersare not necessarily our "yes," and depend on how responsive the cost of insuranceis to the amountspent on protection. Sheltersand seat belts are ways to self-insure, have costs that are and essentially unrelatedto the probability the hazards; therefore of they wouldtendto reduce(perhapsonlyslightly) incentive avoid a war to the or an automobile accident. theotherhand,if thecost in time, On embarrassment, etc. of applyingfor relief,unemployment compensation, or parentalsupportwere sufficiently positively relatedto its frequency, the answersmightwell be "no": the availability insurance of mightencourage theinsured makehis own efforts. to
Appendix A Self-Insurance and Market Insurance If both self-insurance market and insurance available, the expectedutility are is


(1 0

P) U(Iie

c - s ) + p U(Ile -L(Le, c) -c + s). (Al)

The values of c and s that maximizethis function must satisfythe first-order optimality conditions


(1 0

P) U11it pUot_ l+
p) U1' - p Uo'[L'(c*) + 1
8 -

0. (A3)

Clearly, equation (A3) would be satisfiedonly if [L'(c*) + 1] < 0: onlyif expenditures self-insurance on increasedthe net incomein the hazardous state.

644 conditions are optimality Second-order
U*ss(1 -




p) U1" JO + p Uo"1 < 0




p) U1" + p UO"82 -p (U*sc)2 >

Uo'L" <0

(A5) (A6)


U*S8 U*cc


U" Equations (A4) and A5) are obviously satisfiedif everywhere < 0 and = 2LI0c2 > 0, that is, if the marginalutilityof income and the marginal are also These assumptions are of productivity self-insurance both decreasing. to sufficient satisfyequation (A6) since U*Sc
A -

(1 -

) ul"'

P Uo"'' < ?. 1, we can write


conditiont Utilizing first-order the

- p(1 - p) Uo' Ul" L

p2 Uo' Uo"L",

whichis positive if U" < 0 and L" > 0. A. Termsof Trade Effects of The effect an increasein it on theoptimalvalues of s and c-Ile, LC,and p held equations(A2) and (A3) withrespect constant-can be foundby differentiating to it. By Cramer'srule, ds*



-pU A
p( 1

p) Uo'Ul'L" + p( 1


U P i
= < 0

p) Uo'Ul"s*jtL"]

(A8) + (1 - p)U" s* i and - A2 ( - P) Ul s* -(1-p)U1' where -A1 are thepartialderivatives (A2) and (A3) withrespectto it. Similarly, of

dc* dit

A2U*ss - AU*sc

[ A




p)2Ul'Ul"jt -p)U 'U0"]


>0. (A9)

can Hence, marketinsuranceand self-insurance be consideredsubstitutes. of the By similarreasoning, effect an increasein p on the optimalvalues of is s and c, given that 2. in t = [(1 + 2.)p]l(1 - p), Ile and Ile are constant, foundto be

ds* dp










+ (1 -p) Uo'Ul" - pUl'U0"1i + pU0'U0"82]also

< 0; (A10)





B2U*cc - BlU*8C

1 [-



p) U'Uo1"t







where -B1

and m'(P) = anlaP = T/Fp(l -p)].

= (1 - p)Ul"s*jt'(p)it,


B = U1'


U0' + (1 -p)U1's*t'(p),

B. An Endowment Effect
The effect a decreasein I of be shownto be
Ile _

Ie, t, p, and L' held constant-can (+) (+)

ds* 9L -p2U0'U0"IL" As* 02Uot~oo -L L0t A aLe dIoe
where, assumption, by aLlaLe > 0; and



dc* dld10' oe

SPUO8-U*S pPUo" U*8c dL (A13) aLe =~~~~~~~~0, A A&
U*88. If the change in Ioe also

since by equations (A7) and (A8) U*s, =changed the results L', wouldbe different. Appendix B

Self-Protection and Market Insurance If both market is insurance and self-protection available,the expectedutility are - [ 1- p(pe, r) U(I1e U* -r -s 7r(r)) + p(pe, r) U(Ioe -r + s).]


The first-order conditions are optimality

U*, - - (1 O
U*r -

P) Ul

+ pUot_ 0(B2)

p'(r*) ( U

1 - p) U1' [ 1 + s*jt'(r)

p pUo'

wherep'(r*) < 0 and m'(r*)< 0. Second-order conditions that are
U*8 _ (1

0 O (B3)

p) U1"'2 + pUJ' < O


U* rr

- P" (r*) ( U

UO) + (1 - p) U1"[ 1 + s*r'(r*)2



+ 2p'(r*) { U1'[l + s*n'(r)
I U*ssU*rr




p) Ul's*nt (r*) < 0. (B6)








Equations (B4) and (B5) would be satisfiedif U" < 0, if both p"(r*) and a"(r*) > 0, and if D = 2p'(r*){U1'[1 + s*jt'(r*)] - Uo'} (whichis positiveif U" < 0)47 were small in absolute value relativeto the othertermsin equation (B5). These conditions are also sufficient satisfyequation (B6) if, in parto D ticular, < (1 - p)Ul's*jT"(r*) + p"(r*)(Ul - UO). Since it = [(1 + X)p]/(1 -p), the effect an increasein r on it would be of

(1 X) p'(r)
(1 - p)2





where )'(r) gives the effecton the loading of an additional expenditure on self-protection is generally and assumed to be positive.48 0 If insurance were always available at an actuarially fairprice,thenX(P) for all p; hence


p(r) 1-p(r)



p'(r) (1_p)2 2 [p'(r) ]2
( p)3



Equation (B2) reducesto


p"(r) p))2

>0 >

U1' = U0', and equation (B3) to


P)( and 1 + s* jt'(r*) it.


Therefore, 3t'(r*) =-1/[s*


A. Termsof Trade Effects
- p(r*)] If an initially fairpriceit =p(r*)]/Fl wereincreasedby an increase in the loadingwithno changein it'(r),49 the changein the optimalvalues of s and r would be givenby





)S L2_

2 Ul'1

- (1




47 According to equations (B2) and (B9) and the condition U" < 0, U1' [1 + sat' (r*)] < U0' if Jtp/1-P). 48 One can write X'(r) = (ak/ap) (ap/ar, where ap/lr < 0. Hence, X'(r) > 0 only if ak/Op< 0. (But see our discussionin Section IC.) 49 Accordingto equation (B7), an increasein ?i due to an exogenousfactor 0 would not change t' (r) if, and only if, Ep'(r)/(1 - p)] [Ok(r, )/80] =-p [ak' (r, 0)/ 90]. This assumptionis made to separate an autonomous change in the price of insurance from an autonomous change in the effectof self protectionon the price of insurance. 50 Using equations (B8) and (B9) and the second-order optimality conditions



-2(1 P)Ul'U*8r




( )


-CU*S] -

, (B11)







p) Ui"s*t

I8*,r*, p,




U*rw] r*,p, 7r'(r*) constant

p'(r*) U1's* + (


Uj"s* [ 1 + s*rr'(r*)l

(1 -p)U1'-



(fromequation [B9]). Hence, if the price of insuranceincreasedfroman initiallyfairlevel,the demandforboth self-protection market and insurance would decrease. If the price of insurance were always actuariallyfair,the effect an exogeof nous increasein the productivity self-protection s* and r* withno change of on in the endowed probabilities and in the endowed incomes would be given by
ds* D2 U*8r


> ?



dc* _
where D.)

D2 U*88




- p)]1p'l0( and, by assumption ap' 0(3 < 0. TechUl'[s(l nologicalimprovements self-protection thus seen to increasethe demand in are forbothmarket insurance and self-protection.


The Effect ExogenousChangesin p and L of

If insurancewere provided at an actuariallyfair price, and if the endowed probability increaseddue to an exogenousfactory withno changein p'(r), then
discussed above, it follows that U*rr= (1 - p) U1"''2 + pUo0" + 2U1'/s* (1P) (1-p) U1's*rT"' < 0; U*8r (r) U1"2-pU0">0; and

I =





(1 1

p) Ul's* jt"(r)



Since by equation (B4) U*ss < 0, 2Ul'/s* - (1 order for I to be positive.

p) U1' s* JT"(r) must be negativein





ds* = dy



P) U1i"s*TJn i(P)


[U (-)~~~

(1 -p) Ul's*:n (r)

aY where, assumption, by > Op/dy 0, and

(+ )

( B 14)
U*sr] a






since U*_ - - U*',.5l The last resultis intuitively obvioussince a fairprice of insurance impliesthat - p'(r*)= r* of I,); therefore, is independent p providedthatp'(r) is unaffected changesin paY. the same reasoning one by By can showthatan increasein the size of theprospective loss increasesthe optimal values of both s and r. References Arrow,K. J. "Economic Welfare and the Allocationof Resources for Invention." In The Rate and Directionof InventiveActivity:Economic and Social Factors, edited by National Bureau Committee for Economic Research. Princeton, N.J.: Nat. Bur. Econ. Res., 1962. . "Uncertainty and the Welfare Economics of Medical Care." AER 53 (December 1963) :941-73. . "The Role of Securitiesin the Optimal Allocationof Risk Bearing." Rev. Econ. Studies (April 1964) :91-96. . Aspects of the Theory of Risk Bearing. Helsinki: Yrgb Jahnssonin Siitio, 1965. Becker, G. S. "Uncertainty and Insurance,a Few Notes." Unpublished paper, 1968. Becker, G. S., and Michael, R. T. "On the Theory of ConsumerDemand." Unpublished paper, March 1970. Demsetz, H. "Informationand Efficiency: AnotherViewpoint."J. Law and Econ. 12, no. 1 (April 1959): 1-22. Ehrlich,I. "Participationin IllegitimateActivities: An Economic Analysis." Ph.D. dissertation, Columbia Univ., 1970. Hirshleifer, "InvestmentDecision under Uncertainty:Applicationsof the J. State Preference Approach."Q.J.E. 80 (May 1966) :252-77. . Investment, Interestand Capital. EnglewoodCliffs, N.J.: Prentice-Hall, 1970. Lees, D. S., and Rice, R. G. "Uncertainty the WelfareEconomicsof Mediand cal Care: Comment." A.E.R. 55 (March 1965):140-54. Mehr,R. I., and Commack,E. Principlesof Insurance.4th ed. Homewood,Ill.: Irwin, 1966. Mossin,J. "Aspectsof RationalInsurancePurchasing." J.P.E. 76 (July/August): 1968) :553-68. Pratt,J. W. "Risk Aversionin the Small and in the Large." Econometrica32, nos. 1-2 (January-April 1964):122-36. Smith, L. "OptimalInsuranceCoverage."J.P.E. 76 (January/February V. 1968): 68-77.

See equation (B4) and the footnotefollowingequation (B10).


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